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{:[x=(1)/(3)(y+4)],[4x-3y=3(a-x)-2x]:}
In the system of equations,
a is a constant. For what value of
a does the system of linear equations have infinitely many solutions?

$\begin{array}{c} x=\frac{1}{3}(y+4) \\ 4 x-3 y=3(a-x)-2 x \end{array}$ $\newline$ In the system of equations, $a$ is a constant. For what value of $a$ does the system of linear equations have infinitely many solutions?

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Solve rational equations

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Q.

\begin{array}{c} x=\frac{1}{3}(y+4) \\ 4 x-3 y=3(a-x)-2 x \end{array}

\newline

In the system of equations,

a

is a constant. For what value of

a

does the system of linear equations have infinitely many solutions?

Write down the system of equations: First, let's write down the system of equations: $\newline$ $1$ . $x = \frac{1}{3}(y + 4)$ $\newline$ $2$ . $4x - 3y = 3(a - x) - 2x$ $\newline$ For a system of equations to have infinitely many solutions, both equations must represent the same line. This means that the equations must be proportional or one must be a multiple of the other.
Express the first equation in terms of y: Let's express the first equation in terms of y to compare it with the second equation: $\newline$ $x = \frac{1}{3}(y + 4)$ $\newline$ Multiply both sides by $3$ to get rid of the fraction: $\newline$ $3x = y + 4$ $\newline$ Now, let's isolate y: $\newline$ $y = 3x - 4$
Rewrite the second equation to isolate y: Next, we will rewrite the second equation to isolate y as well: $\newline$ $4x - 3y = 3(a - x) - 2x$ $\newline$ First, distribute the $3$ on the right side: $\newline$ $4x - 3y = 3a - 3x - 2x$ $\newline$ Combine like terms on the right side: $\newline$ $4x - 3y = 3a - 5x$ $\newline$ Now, let's isolate y by adding $3$ y to both sides and adding $5$ x to both sides: $\newline$ $4x + 5x = 3y + 3a$ $\newline$ $9x = 3y + 3a$ $\newline$ Divide both sides by $3$ to solve for y: $\newline$ $3x = y + a$ $\newline$ Now, let's isolate y: $\newline$ $y = 3x - a$
Compare the two expressions for y: Now we have two expressions for y: $\newline$ From the first equation: $y = 3x - 4$ $\newline$ From the second equation: $y = 3x - a$ $\newline$ For the system to have infinitely many solutions, these two expressions must be identical, which means that the constants on the right side of each equation must be equal: $\newline$ $-4 = -a$
Solve for a: Solve for a: $\newline$ $a = 4$

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